Curvature and sharp growth rates of log-quasimodes on compact manifolds
Curvature and sharp growth rates of log-quasimodes on compact manifolds
We obtain new optimal estimates for the $L^2(M)\to L^q(M)$, $q\in (2,q_c]$, $q_c=2(n+1)/(n-1)$, operator norms of spectral projection operators associated with spectral windows $[\lambda,\lambda+\delta(\lambda)]$, with $\delta(\lambda)=O((\log\lambda)^{-1})$ on compact Riemannian manifolds $(M,g)$ of dimension $n\ge2$ all of whose sectional curvatures are nonpositive or negative. We show that these two different types of …